Physics 121 Exam Formula Sheet
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Conversion Factors
Mass
1 kg = 1000 g, 1 u = 1.66 10-27 kg
Length
1 m = 100 cm = 3.28 ft = 39.37 in, 1 mi = 1.61 km = 5280 ft = 1609 m
Time
1 day = 86 400 s, 1 yr = 3.156 107s
Speed
60 mi/h = 88.0 ft/s = 26.83 m/s, 100 km/h = 27.78 m/s = 62.14 mi/h
Angle
1 rev = 360° = 2π rad
Force
1 N = 0.225 lb, 1 lb = 4.45 N
Energy
1 eV = 1.60 10-19 J, kcal = 1 Cal = 103 cal = 4.19 kJ,
1kWh = 3.6 106J = 3.6 MJ
Physical Constants
speed of light c 2.998 108m/s
gravitational constant G 6.670 10-11Nm2/kg2
For Earth:
free-fall acceleration g 9.80 m/s2= 32.15 ft/s2
mass ME 5.98 1024 kg
mean radius RE 6370 km
Metric Prefixes
tera- T 1012 giga- G 109 mega- M 106 kilo- k 103
milli- m 10-3 micro- μ 10-6 nano- n 10-9 pico- p 10-12
Chapter 2 - Motion in One Dimension
Position: x(t)x-coordinate at time t, y(t)y-coordinate at time t, etc.
Average speed: average speed total distance moved/Δt (Δt = elapsed time)
Displacement: Δxxfinal – x initial
Average velocity: <v Δx/Δt
Velocity: v dx/dt = slope on x(t) plot. Hence Δx=
Acceleration: a dv/dt = d2v/dt2 = slope on v(t) plot.
Hence Δv =
Kinematic relationships valid iff a = constant:
v = v0 + at, x = x0 + v0t + ½at2, x = x0 + ½ (v + v0)t, v2= v02+ 2a (x - x0)
Free fall:
a = -g (assuming that displacement is taken as positive upward and
ignoring air resistance and other smaller effects)
Chapter 3-Vectors
Vector notation: A is a vector,A is its magnitude (which includes units)and
is a unit vector in the direction of A. Hence The
direction of is given by the right-hand rule.
Vector components in two dimensions: A = Axî+ Ayĵ = (Ax, Ay)
r is a position vector in two dimensions: r = xî+ yĵ = (x, y), r = (x2 + y2)½
Dot (scalar) product: ABAB cos θ(θ is the angle between A and B)
Also
Cross (vector) product: ABABsinθ (is a unit vector normal to
the plane of A and Bin the direction given by the right-hand rule.)
Also .
Chapter 4-Motion in Two [Three] Dimensions
Definitions:
Position: r (x, y, [z]) Displacement: Δr (Δx, Δy, [Δz])
Velocity vdr/dt= (dx/dt, dy/dt, [dz/dt]) = (vx, vy, [vz])
Acceleration: adv/dt= (dvx/dt, dvy/dt, [dvz/dt]) = (ax, ay, [az])
Kinematic relationships valid iff a= constant:
v = v0 + at, r = r0 + v0t + ½ at2, r = r0 + ½ (v + v0)t,
vv=v0v0+ 2aΔr
For free fall with a = (0, -g), v0 = (vx0, vy0) = (v0 cosθ, v0 sin θ)
ax = 0, ay = -g, vx = vx0, vy = vy0 - gt,x = x0 + vx0t, y = y0 + vy0t - ½ gt2
Projectile motion over level terrain with negligible air resistance
Trajectory: y = (tan θ0)x – [g/ (2v02cos2θ0)]x2 (a parabola)
Range: R = (v02/g) sin 2θ0, Maximum height: h = v02sin2θ0/ 2g
Speed v as a function of time: v2 = vx2 + vy2 = vx02 + (vy0-gt)2
Speed v as a function of y: v2 = v02– 2g(y – y0)
Acceleration in Uniform Circular Motion:
ar = v2/ r, directed toward the center of the circle
Radial (centripetal) and Tangential Acceleration: a = ar+ at, with
ar = v2/ r, directed toward the center of the circle,
at = dv/dt, directed toward the direction of motion
Relative Motion: Relative velocity:
vBA = vBC + vCA, and, by induction, vBA = vBC + vCD + vDE + . . . + vXA
Relative acceleration: aBA = aBC, if aCA = 0
Vector Differentiation:
Rectangular form:
Polar form with v = (v, θ):
Chapters 5 and 6 – The Laws of Motion
Newton’s First Law – The First Law of Motion:
In the absence of a force (a free object) moves with a = 0, i.e., if atrest, it
remains at rest. If moving, it continues to move in a straight line at a
constant speed. This is a law describing an inertial reference frame.
If it appears to be violated, the observer’s reference frame is not in an
inertial reference frame.
Definitions:
Acceleration: .
Force: F, a push or pull, the cause of any true acceleration, a vector
quantity. In SI units forces are measured in Newtons (N); in British
units forces are measure in pounds (lbs).
Inertial mass: m, a measure of how an object responds to a force, a scalar
quantity. In SI units masses are measured in kilograms (kg); in British
units masses are measured in slugs.
Newton’s Second Law – The Second Law of Motion:
In an inertial reference frame:
1. Accelerations are caused by forces.
2. a = F/m, or F = ma. The force in this relationship is the net force
acting upon the accelerating body, i.e., it is the sum of all forces
acting upon that body,
3. a is in the same direction as F, always.
Newton’s Third Law – The Third Law of Motion:
If body A exerts a force on body B, then body B exerts a force, equal in
magnitude, but opposite in direction, on body A, i.e.., FAB = FBA, where
FABis the force exerted on body B by body A andFBAis the force exerted
on body A by body B. This law is sometimes called the Law of Action
and Reaction. This is a somewhat misleading title because it implicitly
implies a cause-effect relation between the two forces which are associated
with any interaction. In reality, neither force of a force pair is more
fundamental than the other and neither should be viewed as the cause of
the other. All forces occur in pairs. There are no isolated forces.
Fundamental Forces: There are only four kinds of forces:
1. Gravity: all objects with mass attract all other objects with mass.
2. Electromagnetic Force: all forces experienced at the macroscopic
level, except gravity, are electromagnetic, including tension,
compression, friction, buoyancy and viscous drag.
3. Nuclear Strong Force: this is the force that binds atomic nuclei
together by overcoming the repulsive electromagnetic forces exerted
by the protons on each other. This is a very short-range force.
4. Nuclear Weak Force: this is also a short-range force manifest in
certainly nuclear reactions, including the emission of beta radiation.
Chapter 7 – Energy of a System
Work-Kinetic Energy Theorem: for any objectΔK= WnetWi is
the work done on the object by the ith force acting upon it.
Definitions:
ΔKKfinalKinitial ½ mvf2 ½ mvi2
,Wiis the work done by any force Fi in a displacement
from rito r f. If Fi is (is not) conservative,the integral and thereforeWi
does not (does) depend upon the path taken in evaluating the integral.
Special Cases:
1. F = constant:,
e.g.,
2. One-dimensional motion:
e.g., .
3. W = 0, if .
Power:
Chapter 8 - Conservation of Energy
If a physical quantity X is conserved for an isolated system, then
Conservation of Energy:
For an isolated system:
Definitions:
Conservative force: is a force which does not change the total mechanical
energy of a system when it does work on the system. i.e.,as
a consequence of work done on a system by such a force. For a
conservative force the associated potential energy can be defined byPotential Energy:
,
,
For gravity k = GMm for electrostatic force k = kCQ.
Force from a radial potential energy: .
Chapter 9–Linear Momentum and Collisions
Conservation of Momentum:
For an isolated system(Fext = 0)
Definition of momentum:
For a particle p mv ,
For a system ptotal= Σpi = Σmivi = Mvcm, where MΣmi
(HencevCM)
(Warning: Ktotal is not usually equal to unless the system consists of a
single particle or object.)
2nd Law of Motion and the Impulse-Momentum Theorem:
particle:
system:
(The integral forms of these equations are the impulse-momentum theorems.)
Center of mass:
Chapter 10 – Rigid-body, Fixed-axis Rotation
Dynamics:
Equation of motion:
Kinetic energy:
for a body rolling without slipping
Kinematics:
Angular variables: ,
The translational kinematic and dynamic relationships of other chapters
hold for these variables if we identify t↔t, x↔θ, v↔ω, a↔α, m↔I, F↔τ,
p↔L, K↔K, k↔κ, W↔W, e.g., Newton’s 2nd Law: F= ma ↔ τ= Iα
Angular units: 1 rev = 360° = 2π rad
For constant angular velocity,
Rotational Inertia:
General definition:
Parallel-axis theorem: I = Icm+ Md2
Specific uniform bodies:
Solid sphere about a diameter: Solid cylinder about symmetry axis:
Sph. shell about a diameter: Solid cyl abt centered diam:
Hoop or cyl shell about sym axis: Thin rod, abt centered axis:
Hollow cyl about sym axis: “, abt axis through an end:
Rectangular plate abt axis through center:
Chapter 11 – Torque and Angular Momentum
Conservation of Angular Momentum:
For an isolated system (τext = 0)
Second Law of Motion:
Definition of torque: τr F
particle:
system:
(The integral forms of these equations are the angularimpulse-momentum
theorems.)
Definition of angular momentum:
Rolling (for a object of cylindrical symmetry, rolling without slipping):
and
Chapter 12 – Static Equilibrium, Elasticity
Two equilibrium conditions:
Vector format: (1) , (2)
For coplanar forces: (1a) (1b)
(2a).
If (1) and (2) are true are true about one inertial reference point, they are
true for all inertial reference points.
Center of Gravity:
Note that ifg is constant across the mass distribution (or varies negligibly) then rCG = (or ) rCM.
Chapter 13 Universal Gravitation
Law of Universal Gravitation:
where F21 is the force exerted by m2 on m1, is a unit vector directed from m2 towards m1, i.e., the force is attractive, and
This expression is valid for point masses and also for symmetric spheres (r being the center-to-center separation). Where more than two masses are present, the net force on any particular mass is obtained by simple superposition, i.e., .
Astrophysical consequences:
g of earth
Kepler’s Laws of planetary motion: (1) elliptical (conic section) orbits,
(2) or L = constant, (3)
Motion in binary star systems
Large-scale cosmic structure
Gravitational potential energy:
. Note that U (r= ) = 0.
Bound and unbound orbits. Superposition applies.
Gravitational field:
.
Gravitaional mass versus inertial mass:
mia = mgg, or
.
Any departure of a from g (in a vacuum) would imply the two are different. Experiment verifies that any difference must be <1 part in 1011.